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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risksModeling and covering catastrophic risks Arthur Charpentier AXA Risk College, April 2007 arthur.charpentier@ensae.fr 1
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 2
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 3
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Some stylized facts“climatic risk in numerous branches of industry is more important than the riskof interest rates or foreign exchange risk” (AXA 2004, quoted in Ceres (2004)). Figure 1: Major natural catastrophes (from Munich Re (2006).) 4
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Some stylized facts: natural catastrophesIncludes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail,drought, ﬂoods... Date Loss event Region Overall losses Insured losses Fatalities 25.8.2005 Hurricane Katrina USA 125,000 61,000 1,322 23.8.1992 Hurricane Andrew USA 26,500 17,000 62 17.1.1994 Earthquake Northridge USA 44,000 15,300 61 21.9.2004 Hurricane Ivan USA, Caribbean 23,000 13,000 125 19.10.2005 Hurricane Wilma Mexico, USA 20,000 12,400 42 20.9.2005 Hurricane Rita USA 16,000 12,000 10 11.8.2004 Hurricane Charley USA, Caribbean 18,000 8,000 36 26.9.1991 Typhoon Mireille Japan 10,000 7,000 62 9.9.2004 Hurricane Frances USA, Caribbean 12,000 6,000 39 26.12.1999 Winter storm Lothar Europe 11,500 5,900 110Table 1: The 10 most expensive natural catastrophes, 1950-2005 (from MunichRe (2006)). 5
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Some stylized facts: man-made catastrophesIncludes industry ﬁre, oil & gas explosions, aviation crashes, shipping and raildisasters, mining accidents, collapse of building or bridges, terrorism... Date Location Plant type Event type Loss (property) 23.10.1989 Texas, USA petrochemical∗ vapor cloud explosion 839 04.05.1988 Nevada, USA chemical explosion 383 05.05.1988 Louisiana, USA reﬁnery vapor cloud explosion 368 14.11.1987 Texas, USA petrochemical vapor cloud explosion 282 07.07.1988 North sea platform∗ explosion 1,085 26.08.1992 Gulf of Mexico platform explosion 931 23.08.1991 North sea concrete jacket mechanical damage 474 24.04.1988 Brazil plateform blowout 421Table 2: Onshore and oﬀshore largest property damage losses (from 1970-1999).The largest claim is now the 9/11 terrorist attack, with a US$ 21, 379 millioninsured loss.∗ evaluated loss US$ 2, 155 million and explosion on platform piper Alpha, US$ 3, 409 million (Swiss Re (2006)). 6
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks What is a large claim ?An academic answer ? Teugels (1982) deﬁned “large claims”, Answer 1 “large claims are the upper 10% largest claims”, Answer 2 “large claims are every claim that consumes at least 5% of the sum of claims, or at least 5% of the net premiums”, Answer 3 “large claims are every claim for which the actuary has to go and see one of the chief members of the company”.Examples Traditional types of catastrophes, natural (hurricanes, typhoons,earthquakes, ﬂoods, tornados...), man-made (ﬁres, explosions, businessinterruption...) or new risks (terrorist acts, asteroids, power outages...).From large claims to catastrophe, the diﬀerence is that there is a before thecatastrophe, and an after: something has changed ! 7
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks What is a catastrophe ? Before Katrina After Katrina Figure 2: Allstate’s reinsurance strategies, 2005 and 2006. 8
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The impact of a catastrophe • Property damage: houses, cars and commercial structures, • Human casualties (may not be correlated with economic loss), • Business interruptionExample • Natural Catastrophes - USA: succession of natural events that have hit insurers, reinsurers and the retrocession market • lack of capacity, strong increase in rate • Natural Catastrophes - nonUSA: in Asia (earthquakes, typhoons) and Europe (ﬂood, drought, subsidence) • sui generis protection programs in some countries 9
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The impact of a catastrophe• Storms - Europe: high speed wind in Europe and US, considered as insurable• main risk for P&C insurers• Terrorism, including nuclear, biologic or bacteriologic weapons• lack of capacity, strong social pressure: private/public partnerships• Liabilities, third party damage• growth in indemnities (jurisdictions) yield unsustainable losses• Transportation (maritime and aircrafts), volatile business, and concentrated market 10
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Probabilistic concepts in risk managementLet X1 , ..., Xn denote some claim size (per policy or per event), • the survival probability or exceedance probability is F (x) = P(X > x) = 1 − F (x), • the pure premium or expected value is ∞ ∞ E(X) = xdF (x) = F (x)dx, 0 0 • the Value-at-Risk or quantile function is −1 V aR(X, u) = F −1 (u) = F (1 − u) i.e. P(X > V aR(X, u)) = 1 − u, • the return period is T (u) = 1/F (x)(u). 11
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The density of the exponential distribution The exceedance distribution 0.5 1.0 0.4 0.8 mean = 1 mean = 2 mean = 5 mean = 1 0.3 0.6 mean = 2 Probability mean = 5 0.2 0.4 0.1 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Claim size Claim size The quantile function of the exponential distribution The return period function 15 12 10 10 8 mean = 1 Claim size Claim size mean = 2 mean = 5 6 5 4 mean = 1 2 mean = 2 mean = 5 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 Probability level Time Figure 3: Probabilistic concepts, case of exponential claims. 12
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Modeling catastrophes• Man-made catastrophes: modeling very large claims,• extreme value theory (ex: business interruption)• Natural Catastrophes: modeling very large claims taking into accont accumulation and global warming• extreme value theory for losses (ex: hurricanes)• time series theory for occurrence (ex: hurricanes)• credit risk models for contagion or accumulation 13
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Updating actuarial modelsIn classical actuarial models (from Cramér and Lundberg), one usuallyconsider • a model for the claims occurrence, e.g. a Poisson process, • a model for the claim size, e.g. a exponential, Weibull, lognormal...For light tailed risk, Cramér-Lundberg’s theory gives a bound for the ruinprobability, assuming that claim size is not to large. Furthermore, additionalcapital to ensure solvency (non-ruin) can be obtained using the central limittheorem (see e.g. RBC approach). But the variance has to be ﬁnite.In the case of large risks or catastrophes, claim size has heavy tails (e.g. thevariance is usually inﬁnite), but the Poisson assumption for occurrence is stillrelevant. 14
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Updating actuarial models NExample For business interruption, the total loss is S = Xi where N is i=1Poisson, and the Xi ’s are i.i.d. Pareto.Example In the case of natural catastrophes, claim size is not necessarily huge,but the is an accumulation of claims, and the Poisson distribution is not relevant.But if considering events instead of claims, the Poisson model can be relevant.But the Poisson process is nonhomogeneous. NExample For hurricanes or winterstorms, the total loss is S = Xi where N is i=1 NiPoisson, and Xi = Xi,j , where the Xi,j ’s are i.i.d. j=1 15
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 16
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Some empirical facts about business interruptionBusiness interruption claims can be very expensive. Zajdenweber (2001)claimed that it is a noninsurable risk since the pure premium is (theoretically)inﬁnite.Remark For the 9/11 terrorist attacks, business interruption represented US$ 11billion. 17
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Some results from Extreme Value TheoryWhen modeling large claims (industrial ﬁre, business interruption,...): extremevalue theory framework is necessary.The Pareto distribution appears naturally when modeling observations over agiven threshold, b x F (x) = P(X ≤ x) = 1 − , where x0 = exp(−a/b) x0Then equivalently log(1 − F (x)) ∼ a + b log x, i.e. for all i = 1, ..., n, log(1 − Fn (Xi )) ∼ a + b · log Xi .Remark: if −b ≥ 1, then EP (X) = ∞, the pure premium is inﬁnite.The estimation of b is a crucial issue. 18
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Cumulative distribution function, with confidence interval 1.0 log−log Pareto plot, with confidence interval 0 logarithm of the survival probabilities −1 0.8 cumulative probabilities −2 0.6 −3 0.4 −4 0.2 −5 0.0 0 1 2 3 4 5 0 1 2 3 4 5 logarithm of the losses logarithm of the losses Figure 4: Pareto modeling for business interruption claims. 19
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Why the Pareto distribution ? historical perspectiveVilfredo Pareto observed that 20% of the population owns 80% of the wealth. 80% of the claims 20% of the losses 20% of the claims 80% of the losses Figure 5: The 80-20 Pareto principle.Example Over the period 1992-2000 in business interruption claims in France,0.1% of the claims represent 10% of the total loss. 20% of the claims represent73% of the losses. 20
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Why the Pareto distribution ? historical perspective Lorenz curve of business interruption claims 1.0 0.8 73% OF Proportion of claim size THE LOSSES 0.6 0.4 20% OF 0.2 THE CLAIMS 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of claims number Figure 6: The 80-20 Pareto principle. 21
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Why the Pareto distribution ? mathematical explanationWe consider here the exceedance distribution, i.e. the distribution of X − u giventhat X > u, with survival distribution G(·) deﬁned as F (x + u) G(x) = P(X − u > x|X > u) = F (u)This is closely related to some regular variation property, and only powerfunction my appear as limit when u → ∞: G(·) is necessarily a power function. The Pareto model in actuarial literatureSwiss Re highlighted the importance of the Pareto distribution in two technicalbrochures the Pareto model in property reinsurance and estimating propertyexcess of loss risk premium: The Pareto model.Actually, we will see that the Pareto model gives much more than only apremium. 22
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Large claims and the Pareto modelThe theorem of Pickands-Balkema-de Haan states that if the X1 , ..., Xn areindependent and identically distributed, for u large enough, −1/ξ 1+ξ x if ξ = 0, P(X − u > x|X > u) ∼ Hξ,σ(u) (x) = σ(u) exp − x if ξ = 0, σ(u)for some σ(·). It simply means that large claims can always be modeled using the(generalized) Pareto distribution.The practical question which always arises is then “what are large claims”, i.e.how to chose u ? 23
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks How to deﬁne large claims ? • Use of the k largest claims: Hill’s estimatorThe intuitive idea is to ﬁt a linear straight line since for the largest claimsi = 1, ..., n, log(1 − Fn (Xi )) ∼ a + blog Xi . Let bk denote the estimator based onthe k largest claims.Let {Xn−k+1:n , ..., Xn−1:n , Xn:n } denote the set of the k largest claims. Recallthat ξ ∼ −1/b, and then n 1 ξ= log(Xn−k+i:n ) − log(Xn−k:n ). k i=1 24
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks 2.5 Hill estimator of the slope Hill estimator of the 95% VaR 10 2.0 8 quantile (95%) slope (−b) 6 1.5 4 1.0 2 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Figure 7: Pareto modeling for business interruption claims: tail index. 25
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks • Use of the claims exceeding u: maximum likelihoodA natural idea is to ﬁt a generalized Pareto distribution for claims exceeding u,for some u large enough.threshold [1] 3, we chose u = 3p.less.thresh [1] 0.9271357, i.e. we keep to 8.5% largest claimsn.exceed [1] 87method [1] “ml”, we use the maximum likelihood technique,par.ests, we get estimators ξ and σ, xi sigma 0.6179447 2.0453168par.ses, with the following standard errors xi sigma 0.1769205 0.4008392 26
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks 5.0 MLE of the tail index, using Generalized Pareto Model Estimation of VaR and TVaR (95%) 5 e−02 1 e−02 4.5 1−F(x) (on log scale) 95 tail index 2 e−03 4.0 99 5 e−04 3.5 1 e−04 3.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 5 10 20 50 100 200 x (on log scale)Figure 8: Pareto modeling for business interruption claims: VaR and TVaR. 27
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks From the statistical model of claims to the pure premiumConsider the following excess-of-loss treaty, with a priority d = 20, and an upperlimit 70. Historical business interruption claims 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1993 1994 1995 1996 1997 1998 1999 2000 2001 Figure 9: Pricing of a reinsurance layer. 28
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks From the statistical model of claims to the pure premiumThe average number of claims per year is 145, year 1992 1993 1994 1995 1996 1997 1998 1999 2000 frequency 173 152 146 131 158 138 120 156 136 Table 3: Number of business interruption claims. 29
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks From the statistical model of claims to the pure premiumFor a claim size x, the reinsurer’s indemnity is I(x) = min{u, max{0, x − d}}.The average indemnity of the reinsurance can be obtained using the Paretomodel, ∞ u E(I(X)) = I(x)dF (x) = (x − d)dF (x) + u(1 − F (u)), 0 dwhere F is a Pareto distribution.Here E(I(X)) = 0.145. The empirical estimate (burning cost) is 0.14.The pure premium of the reinsurance treaty is 20.6.Example If d = 50 and d = 50, π = 8.9 (12 for burning cost... based on 1 claim). 30
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 31
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Figure 10: Hurricanes from 2001 to 2004. 32
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Figure 11: Hurricanes 2005, the record year. 33
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Increased value at riskIn 1950, 30% of the world’s population (2.5 billion people) lived in cities. In2000, 50% of the world’s population (6 billon).In 1950 the only city with more than 10 million inhabitants was New York.There were 12 in 1990, and 26 are expected by 2015, including • Tokyo (29 million), • New York (18 million), • Los Angeles (14 million). • Increasing value at risk (for all risks)The total value of insured costal exposure in 2004 was • $1, 937 billion in Florida (18 million), • $1, 902 billion in New York. 34
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Two techniques to model large risks • The actuarial-statistical technique: modeling historical series,The actuary models the occurrence process of events, and model the claim size(of the total event).This is simple but relies on stability assumptions. If not, one should modelchanges in the occurrence process, and should take into account inﬂation orincrease in value-at-risk. • The meteorological-engineering technique: modeling natural hazard and exposure.This approach needs a lot of data and information so generate scenarios takingall the policies speciﬁcities. Not very ﬂexible to estimate return periods, andworks as a black box. Very hard to assess any conﬁdence levels. 35
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The actuarial-statistical approach • Modeling event occurrence, the problem of global warming.Global warming has an impact on climate related hazard (droughts, subsidence,hurricanes, winterstorms, tornados, ﬂoods, coastal ﬂoods) but not geophysical(earthquakes). • Modeling claim size, the problem of increase of value at risk and inﬂation.Pielke & Landsea (1998) normalized losses due to hurricanes, using bothpopulation and wealth increases, “with this normalization, the trend of increasingdamage amounts in recent decades disappears”. 36
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Impact of global warming on natural hazard Number of hurricanes, per year 1851−2006 25 Frequency of hurricanes 20 15 10 5 0 1850 1900 1950 2000 Year Figure 12: Number of hurricanes and major hurricanes per year. 37
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks More natural hazards with higher value at riskConsider the example of tornados. Number of tornados in the US, per month 400 300 Number of tornados 200 100 0 1960 1970 1980 1990 2000 Year Figure 13: Number of tornadoes (from http://www.spc.noaa.gov/archive/). 38
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks More natural hazards with higher value at riskThe number of tornados per year is (linearly) lincreasing. Distribution of the number of tornados, per year (1960, 1980, 2000) 0.05 0.04 0.03 0.02 0.01 0.00 40 60 80 100 120 140 160 Figure 14: Evolution of the distribution of the number of tornados per year. 39
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks More natural hazards with higher value at risk Return period for tornados: more natural hazard 50 40 Claim size 30 HOMOTHETIC TRANSFORMATION DUE TO 20 MORE NATURAL HASARD PER YEAR 10 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 100 200 300 400 500 600 700 800 900 1000 0 0 20 40 60 80 100 Time (in years) Figure 15: Impact of global warming on the return period. 40
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks More natural hazards with higher value at riskThe most damaging tornadoes in the U.S. (1890-1999), adjusted with wealth, arethe following, Date Location Adjusted loss 28.05.1896 Saint Louis, IL 2,916 29.09.1927 Saint Louis, IL 1,797 18.04.1925 3 states (MO, IL, IN) 1,392 10.05.1979 Wichita Falls, TX 1,141 09.06.1953 Worcester, MA 1,140 06.05.1975 Omaha, NE 1,127 08.06.1966 Topeka, KS 1,126 06.05.1936 Gainesville, GA 1,111 11.05.1970 Lubbock, TX 1,081 28.06.1924 Lorain-Sandusky, OH 1,023 03.05.1999 Oklahoma City, OK 909 11.05.1953 Waco, TX 899 27.04.1890 Louisville, KY 836 Table 4: Most damaging tornadoes (from Brooks & Doswell (2001)). 41
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks More natural hazards with higher value at risk Return period for tornados: more value at risk 50 40 Claim size 30 20 HOMOTHETIC TRANSFORMATION DUE TO THE INCREASE OF VALUE AT RISK 10 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 100 200 300 400 500 600 700 800 900 1000 0 0 20 40 60 80 100 Time (in years) Figure 16: Impact of increase of value at risk on the return period. 42
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Cat models: the meteorological-engineering approachThe basic framework is the following, 1. the natural hazard model: generate stochastic climate scenarios, and assess perils, 2. the engineering model : based on the exposure, the values, the building, calculate damage, 3. the insurance model: quantify ﬁnancial losses based on deductibles, reinsurance (or retrocession) treaties. 43
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida Figure 17: Florida and Hurricanes risk. 44
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida1. the natural hazard model: generate stochastic climate scenarios, and assess perils, Figure 18: Generating stochastic climate scenarios. 45
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida1. the natural hazard model: generate stochastic climate scenarios, and assess perils, Figure 19: Generating stochastic climate scenarios. 46
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida1. the natural hazard model: generate stochastic climate scenarios, and assess perils, Figure 20: Checking outputs of climate scenarios. 47
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida1. the natural hazard model: generate stochastic climate scenarios, and assess perils, Figure 21: Checking outputs of climate scenarios. 48
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida2. the engineering model : based on the exposure, the values, the building, calculate damage, Figure 22: Modeling the vulnerability. 49
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical example: Hurricanes in Florida2. the engineering model : based on the exposure, the values, the building, calculate damage, Figure 23: Modeling the vulnerability. 50
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Hurricanes in Florida: Rare and extremal events ?Note that for the probabilities/return periods of hurricanes related to insuredlosses in Florida are the following (source: Wharton Risk Center & RMS) $ 1 bn $ 2 bn $ 5 bn $ 10 bn $ 20 bn $ 50 bn 42.5% 35.9% 24.5% 15.0% 6.9% 1.7% 2 years 3 years 4 years 7 years 14 years 60 years $ 75 bn $ 100 bn $ 150 bn $ 200 bn $ 250 bn 0.81% 0.41% 0.11% 0.03% 0.005% 123 years 243 years 357 years 909 years 2, 000 years Table 5: Extremal insured losses (from Wharton Risk Center & RMS).Recall that historical default (yearly) probabilities are AAA AA A BBB BB B 0.00% 0.01% 0.05% 0.37% 1.45% 6.59% - 10, 000 years 2, 000 years 270 years 69 years 15 years Table 6: Return period of default (from S&P’s (1981-2003)). 51
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Are there any safe place to be ? Figure 24: Looking for a safe place ? going in North-East...? 52
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks A practical case: North-East Hurricanes in the U.S. Figure 25: North-East Hurricanes in the U.S.: the 1938 hurricane 53
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks North-East Hurricanes: the 1938 experience• Peak Steady Winds - 186 mph at Blue Hill Observatory, MA.• Lowest Pressure - 946.2 mb at Bellport, NY• Peak Storm Surge - 17 ft. above normal high tide• Peak Wave Heights - 50 ft. at Gloucester, MA• Deaths 700 (600 in New England)• Homeless 63,000• Homes, Buildings Destroyed 8,900• Boats Lost 3,300• Trees Destroyed - 2 Billion (approx.)• Cost US$ 300 million (24 billion - 2005 adjusted) 54
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks North-East Hurricanes: further (recent) experience1938 New England Hurricane, Cat 51954 Carol, Cat 3 (Rhode Island, Connecticut, Massachusetts)1954 Edna, Cat 3 (North Carolina, Massachusetts, New Hampshire, Maine)1960 Donna, Cat 5 (New York, Rhode Island, Connecticut, Massachusetts)1961 Esther, Cat 4 (Massachusetts, New Jersey, New York, New Hampshire)1985 Gloria, Cat 4 (Virginia, New York, Connecticut)1991 Bob, Cat 3 (Rhode Island, Massachusetts)1996 Bertha, Cat 3 (North Carolina)1999 Floyd, Cat 4 (North Carolina, Virginia, Delaware, Pennsylvania, New Jersey, New York, Vermont, Maine)2003 Isabel, Cat 4 (North Carolina, Virginia, Washington D.C., Delaware)2004 Charley, Cat 4 (Rhode Island, Virginia, North Carolina) 55
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks North-East Hurricanes: probabilities and return periodAccording to the United States Landfalling Hurricane Probability Project, • 21% probability that NY City/Long Island will be hit with a tropical storm or hurricane in 2007, • 6% probability that NY City/Long Island will be hit with a major hurricane (category 3 or more) in 2007, • 99% probability that NY City/Long Island will be hit with a tropical storm or hurricane in the next 50 years. • 26% probability that NY City/Long Island will be hit with a major hurricane (category 3 or more) in the next 50 years. 56
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks North-East Hurricanes: potential lossesFigure 26: Coast risk in the U.S. and the nightmare scenario in New Jersey (US$100 billion). 57
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Modelling contagion in credit risk models cat insurance credit risk n total number of insured n number of credit issuers 1 if policy i claims 1 if issuers i defaults Ii = Ii = 0 if not 0 if not Mi total sum insured Mi nominal Xi exposure rate 1 − Xi recovery rate 58
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Modelling contagion in credit risk modelsIn CreditMetrics, the idea is to generate random scenario to get the Proﬁt &Loss distribution of the portfolio. • the recovery rate is modeled using a beta distribution, • the exposure rate is modeled using a MBBEFD distribution (see Bernegger (1999)).To generate joint defaults, CreditMetrics proposed a probit model. 59
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The case of ﬂoodFigure 27: August 2002 ﬂoods in Europe, ﬂood damage function, (Munich Re(2006)). 60
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The case of ﬂood Figure 28: Paris, 1910, the centennial ﬂood. 61
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Assessing return period in a changing environment ? Figure 29: Hydrological scheme of the Seine. 62
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Assessing return period in a changing environment ? Figure 30: Hydrological scheme of the Seine. 63
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Comparison of the two approachesFFFFFFFF 64
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 65
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Risk management solutions ?• Equity holding: holding in solvency margin+ easy and basic buﬀer− very expensive• Reinsurance and retrocession: transfer of the large risks to better diversiﬁed companies+ easy to structure, indemnity based− business cycle inﬂuences capacities, default risk• Side cars: dedicated reinsurance vehicules, with quota share covers+ add new capacity, allows for regulatory capital relief− short maturity, possible adverse selection 66
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Risk management solutions ?• Industry loss warranties (ILW) : index based reinsurance triggers+ simple to structure, no credit risk− limited number of capacity providers, noncorrelation risk, shortage of capacity• Cat bonds: bonds with capital and/or interest at risk when a speciﬁed trigger is reached+ large capacities, no credit risk, multi year contracts− more and more industry/parametric based, structuration costs 67
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Insured losses SELF PRIMARY INSURANCE SIDE CARS INSURANCE REINSURANCE ILW CAR BONDS 0.04 0.03 Probability density 0.02 0.01 0.00 DEDUCTIBLE 0 20 40 60 80 100 Claim losses Figure 31: Risk management solutions for diﬀerent types of losses. 68
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Additional capital, post−Katrina reinsurance market 2.5 BN$ 27 BN$ 4 BN$ ADDITIONAL EQUITY 3.5 BN$ 8 BN$ INSURANCE LINKED SECURITIES 9 BN$ EXISTING START UP SIDE ILW CAT TOTAL COMPANIES CARS BONDS Figure 32: Risk management solutions for diﬀerent types of losses. 69
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Retrocession market, 1998−2006 17155 ILW Retrocession market (including ILW) Side cars capital capital markets 12505 Cat bonds issuances 7452 6561 4576 3717 3447 3171 2272 1998 1999 2000 2001 2002 2003 2004 2005 2006 Figure 33: Capital market provide half of the retrocession market. 70
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Trigger deﬁnition for peak risk• indemnity trigger: directly connected to the experienced damage+ no risk for the cedant, only one considered by some regulator (NAIC)− time necessity to estimate actual damage, possible adverse selection (audit needed)• industry based index trigger: connected to the accumulated loss of the industry (PCS)+ simple to use, no moral hazard− noncorrelation risk 71
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Trigger deﬁnition for peak risk• environmental based index trigger: connected to some climate index (rainfall, windspeed, Richter scale...) measured by national authorities and meteorological oﬃces+ simple to use, no moral hazard− noncorrelation risk, related only to physical features (not ﬁnancial consequences)• parametric trigger: a loss event is given by a cat-software, using climate inputs, and exposure data+ few risk for the cedant if the model ﬁts well− appears as a black-box 72
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Figure 34: Actual losses versus payout (cat option). 73
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Reinsurance The insurance approach (XL treaty) 35 30 25 REINSURER Loss per event 20 15 INSURER 10 INSURED 5 0 0.0 0.2 0.4 0.6 0.8 1.0 Event Figure 35: The XL reinsurance treaty mechanism. 74
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Group net W.P. net W.P. loss ratio total Shareholders’ Funds (2005) (2004) (2005) (2004) Munich Re 17.6 20.5 84.66% 24.3 24.4 Swiss Re (1) 16.5 20 85.78% 15.5 16 Berkshire Hathaway Re 7.8 8.2 91.48% 40.9 37.8 Hannover Re 7.1 7.8 85.66% 2.9 3.2 GE Insurance Solutions 5.2 6.3 164.51% 6.4 6.4 Lloyd’s 5.1 4.9 103.2% XL Re 3.9 3.2 99.72% Everest Re 3 3.5 93.97% 3.2 2.8 Reinsurance Group of America Inc. 3 2.6 1.9 1.7 PartnerRe 2.8 3 86.97% 2.4 2.6 Transatlantic Holdings Inc. 2.7 2.9 84.99% 1.9 2 Tokio Marine 2.1 2.6 26.9 23.9 Scor 2 2.5 74.08% 1.5 1.4 Odyssey Re 1.7 1.8 90.54% 1.2 1.2 Korean Re 1.5 1.3 69.66% 0.5 0.4 Scottish Re Group Ltd. 1.5 0.4 0.9 0.6 Converium 1.4 2.9 75.31% 1.2 1.3 Sompo Japan Insurance Inc. 1.4 1.6 25.3% 15.3 12.1 Transamerica Re (Aegon) 1.3 0.7 5.5 5.7 Platinum Underwriters Holdings 1.3 1.2 87.64% 1.2 0.8 Mitsui Sumitomo Insurance 1.3 1.5 63.18% 16.3 14.1Table 7: Top 25 Global Reinsurance Groups in 2005 (from Swiss Re (2006)). 75
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Side carsA hedge fund that wishes to get into the reinsurance business will start a specialpurpose vehicle with a reinsurer The hedge fund is able to get into reinsurancewithout Hiring underwriters Buying models Getting rated by the rating agencies 76
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks ILW - Insurance Loss WarrantyIndustry loss warranties pay a ﬁxed amount based of the amount of industry loss(PCS or SIGMA).Example For example, a $30 million ILW with a $5 billion trigger. Cat bonds and securitizationBonds issued to cover catastrophe risk were developed subsequent to HurricaneAndrewThese bonds are structured so that the investor has a good return if there are noqualifying events and a poor return if a loss occurs. Losses can be triggered on anindustry index or on an indemnity basis. 77
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Cat bonds and securitization The securitization approach (Cat bond) 35 INVESTORS 30 25 SPV Loss per event 20 15 INSURER 10 INSURED 5 0 0.0 0.2 0.4 0.6 0.8 1.0 Event Figure 36: The securitization mechanism, parametric triggered cat bond. 78
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Capital structure, Residential Re, 2001 USAA retention of traditional reinsurance USAA annual US$ 1.6 0.41% exceedance billion probability Residential Re Traditional reinsurance US$ 150 million US$ 300 million USAA part of part of US$ 500 million US$ 500 million annual US$ 1.1 1.12% exceedance billion probability Traditional reinsurance US$ 360 million USAA part of US$ 400 million USAA retention & Florida hurricane catastrophe fund or traditional reinsurance Figure 37: Some cat bonds issued: Residential Re. 79
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Capital structure, Redwood Capital I Ltd, 2001 PCS industry annual losses exceedence US$ probability (billion) 100% 31.5 0.34% 88.9% 30.5 0.37% 77.8% 29.5 0.40% 66.7% 28.5 0.44% 55.6% 27.5 0.48% 44.4% 26.5 0.52% 33.3% 25.5 0.56% 22.2% 24.5 0.61% 11.1% 23.5 0.66% Figure 38: Some cat bonds issued: Redwood Capital. 80
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Capital structure, Atlas Re II, 2001 Traditional retrocession and retention by SCOR Atlas Re II retrocessional agreement, US$ 150 million per event Class A notes, US$ 50 million annual 0.07% exceedance probability Atlas Re II retrocessional agreement, US$ 150 million per event Class B notes, US$ 100 million annual 1.33% exceedance probability Traditional retrocession and retention by SCOR Figure 39: Some cat bonds issued: Redwood Capital. 81
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Property Catastrophe Risk Linked Securities, 2001 600 FRENCH WIND TOKYO EARTHQUAKE CALIFORNIA EARTHQUAKE US S.E. WIND US N.E. WIND 500 SECOND EVENT EUROPEAN WIND 400 JAPANESE EARTHQUAKE MONACO EARTHQUAKE 300 MADRID EARTHQUAKE 200 100 0 Figure 40: Distribution of US$ ar risk, per peril. 82
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Cat bonds versus (traditional) reinsurance: the price• A regression model (Lane (2000))• A regression model (Major & Kreps (2002)) 83
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Figure 41: Reinsurance (pure premium) versus cat bond prices. 84
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Cat bonds versus (traditional) reinsurance: the price • Using distorted premiums (Wang (2000,2002))If F (x) = P(X > x) denotes the losses survival distribution, the pure premium is ∞π(X) = E(X) = 0 F (x)dx. The distorted premium is ∞ πg (X) = g(F (x))dx, 0where g : [0, 1] → [0, 1] is increasing, with g(0) = 0 and g(1) = 1.Example The proportional hazards (PH) transform is obtained when g is apower function.Wang (2000) proposed the following transformation, g(·) = Φ(Φ−1 (F (·)) + λ),where Φ is the N (0, 1) cdf, and λ is the “market price of risk”, i.e. the Sharperatio. More generally, consider g(·) = tκ (t−1 (F (·)) + λ), where tκ is the Student t κcdf with κ degrees of freedom. 85
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Property Catastrophe Risk Linked Securities, 2001 16 Yield spread (%) Lane model Wang model Empirical 14 12 10 8 6 4 2 0 Mosaic 2A Mosaic 2B Halyard Re Domestic Re Concentric Re Juno Re Residential Re Kelvin 1st event Kelvin 2nd event Gold Eagle A Gold Eagle B Namazu Re Atlas Re A Atlas Re B Atlas Re C Seismic Ltd Figure 42: Cat bonds yield spreads, empirical versus models. 86
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Who might buy cat bonds ?In 2004, • 40% of the total amount has been bought by mutual funds, • 33% of the total amount has been bought by cat funds, • 15% of the total amount has been bought by hedge funds.Opportunity to diversify asset management (theoretical low correlation withother asset classes), opportunity to gain Sharpe ratios through cat bonds excessspread. 87
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risksInsure against natural catastrophes and make money ? Return On Equity, US P&C insurers 15 KATRINA RITA WILMA 10 4 hurricanes NORTHRIDGE 5 ANDREW 0 9/11 1990 1995 2000 2005 Figure 43: ROE for P&C US insurance companies. 88
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risksReinsure against natural catastrophes and make money ? Combined Ratio Reinsurance vs. P/C Industry 162.4 160 150 9/11 2004/2005 140 ANDREW HURRICANES 129 130 126.5 125.8 124.6 119.2 120 115.8 115.8 114.3 113.6 110.5 110.1 110.1 111 108.8 108.5 107.4 106.9 106.7 110 108 106.5 105.9 104.8 106 105 101.9 100.9 100.8 100.5 98.3 100 90 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Figure 44: Combined Ratio for P&C US companies versus reinsurance. 89
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 90
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risksSolvency margins when insuring again natural catastrophesWithin an homogeneous portfolios (Xi identically distributed), suﬃciently large X1 + ... + Xn(n → ∞), → E(X). If the variance is ﬁnite, we can also derive a nconﬁdence interval (solvency requirement), if the Xi ’s are independent, n √ Xi ∈ nE(X) ± 2 nVar(X) with probability 99%. i=1 premium risk based capital needNonindependence implies more volatility and therefore more capital requirement. 91
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Implications for risk capital requirements 0.04 0.03 Probability density 99.6% quantile Risk−based capital need 0.02 99.6% quantile Risk−based capital need 0.01 0.00 0 20 40 60 80 100 Annual lossesFigure 45: Independent versus non-independent claims, and capital requirements. 92
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The premium as a fair pricePascal and Fermat in the XVIIIth century proposed to evaluate the “produitscalaire des probabilités et des gains”, n < p, x >= pi xi = EP (X), i=1based on the “règle des parties”.For Quételet, the expected value was, in the context of insurance, the price thatguarantees a ﬁnancial equilibrium. 93
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks What is probability P ?“my dwelling is insured for $ 250,000. My additional premium for earthquakeinsurance is $ 768 (per year). My earthquake deductible is $ 43,750... The more Ilook to this, the more it seems that my chances of having a covered loss are aboutzero. I’m paying $ 768 for this ? ” (Business Insurance, 2001). • Estimated annualized proability in Seatle 1/250 = 0.4%, • Actuarial probability 768/(250, 000 − 43, 750) ∼ 0.37%The probability for an actuary is 0.37% (closed to the actual estimatedprobability), but it is much smaller for anyone else. 94
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The short memory puzzle Percentage of California Homeowners with Earthquake Insurance 32.9 33 33.2 19.5 17.4 16.8 15.7 15.8 14.6 13.3 13.8 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Figure 46: Trajectory of major hurricanes, in 1999 and 2005. 95
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Rational behavior of insurers ?Between September 2004 and September 2005, the real estate prices (MiamiDade county) increased of +45%, despite the 4 hurricanes in 2004. Flyods Hurricane, 1999 The 2005 hurricanes of level 5 64.82 64.82 74.08 83.34 83.34 92.6 92.6 111.12 129.64 166.68 166.68 175.94 185.2 203.72 212.98 203.72 194.46 194.46 212.98 231.5 250.02 250.02 231.5 212.98 194.46 175.94 157.42 166.68 175.94 175.94 148.16 Figure 47: Trajectory of major hurricanes, in 1999 and 2005. 96
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks von Neumann & Morgenstern: expected utility approach Ru (X) = u(x)dP = P(u(X) > x))dxwhere u : [0, ∞) → [0, ∞) is a utility function.Example with an exponential utility, u(x) = [1 − e−αx ]/α, 1 Ru (X) = log EP (eαX ) . αMusiela & Zariphopoulou (2001) used this premium to price derivatives inincomplete markets. 97
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Yaari: distorted utility approach Rg (X) = xdg ◦ P = g(P(X > x))dxwhere g : [0, 1] → [0, 1] is a distorted function.Example if g(x) = I(X ≥ α) Rg (X) = V aR(X, α), and if g(x) = min{x/α, 1}Rg (X) = T V aR(X, α) (also called expected shortfall),Rg (X) = EP (X|X > V aR(X, α)). 98
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Calcul de l’esperance mathématique 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 Figure 48: Expected value xdFX (x) = P(X > x)dx. 99
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Calcul de l’esperance d’utilité 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 Figure 49: Expected utility u(x)dFX (x). 100
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Calcul de l’intégrale de Choquet 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 Figure 50: Distorted probabilities g(P(X > x))dx. 101
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Value-at-Risk and Expected ShortfallThe Value-at-Risk is simply the quantile of a proﬁt & loss distribution, V aR(X, p) = xp = F −1 (p) = sup{x ∈ R, F (x) ≥ p}.Remark This notion is closely related to the return period and ruin probabilities.The Expected Shortfall, or Tail Value-at-Risk, is the expected value above theVaR, T V aR(X, p) = E(X|X > V aR(X, p)). 102
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Worst-case scenariosConsider a set of scenarios, i.e. possible probabilities Q. Consider R(X) = sup {EQ (X)} , Q∈Qthe worst case scenarios pure premium. 103
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risksCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 104
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Coherent risk measuresA risk measure is said to be coherent (from Artzner, Delbaen, Eber &Heath (1999)) if • R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ), • R(·) is positively homogeneous, i.e. for any λ ≤ 0, R(λX) = λR(X), • R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ, • R(·) is subadditive, i.e. R(X + Y ) ≤ R(X) + R(Y ).“subadditivity” can be interpreted as “diversiﬁcation does not increase risk”.Example: the Expected-Shortfall is coherent, the Value-at-Risk is not. 105
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Convex risk measuresA risk measure is said to be convex (from Artzner, Delbaen, Eber & Heath(1999)) if • R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ), • R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ, • R(·) is convex, i.e. R(λX + (1 − λ)Y ) ≤ λR(X) + (1 − λ)R(Y ), for any λ ∈ [0, 1].Hence, if a convex measure satisﬁes the homogeneity condition, it is coherent. 106
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Convex risk measures and Expected-ShortfallAll distortion convex risk measure is a mixture of expected shorfalls, 1 1 −1 R(X) = FX (1 − p)dg(p) = ES(X, 1 − p)dµ(p) = E(ES(X, Θ)), 0 0where Θ is a random variable with values in [0, 1] (Inui & Kijima (2004)). 107
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Estimating a Value-at-RiskA natural idea is to use the Pareto approximation for claims exceeding thresholdu. nLet Nu denote the number of claims exceeding u, Nu = I(Xi > u). i=1If x > u, F (x) = P(X > x) = P(X > u)P(X > x|X > u) = F (u)P(X > x|X > u),where P(X > x|X > u) = G(x)(x − u) is Pareto distributed G(t) = P(X − u ≤ t|X > u) ∼ Hξ,σ (t). 108
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Estimating a Value-at-RiskThus, a natural estimator for F (x) uses a natural estimator for F (u), and thePareto approximation for F u (x), i.e. −1/ξ Nu x−u F (x) = 1 − 1+ξ n σfor all x > u, and u large enough.Thus, a natural estimator for V aR(X, p) is −ξ σ n V aRu (X, p) = u + (1 − p) −1 . ξ Nu 109
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Estimating a Value-at-RiskNote that Hill’s estimator can also be used Hill n −ξn,k V aRk (X, p) = Xn−k:n (1 − p) , kfor some k such that p > 1 − k/n. This estimator can be written Hill n −ξn,k V aRk (X, p) = Xn−k:n + Xn−k:n (1 − p) −1 . k 110
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Estimating an Expected ShortfallSimilarly, the expected shortfall can be estimated simply, 1 σ − ξu ES u (X, p) = V aRu (X, p) · + . 1−ξ [1 − ξ · V aRu (X, p)] 111
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Risk measures for business interruption Lower CI Estimate Upper CI u = 0.5 Var(99.9%) 27.50320 46.57878 99.60774 TVar(99.9%) 48.80638 115.46481 205.12015 u=1 Var(99.9%) 29.09757 47.26090 89.10651 TVaR(99.9%) 56.36115 120.26600 205.12015 u=5 Var(99.9%) 25.73079 43.85056 144.09078 TVar(99.9%) 42.10588 238.93939 205.12015Table 8: Distorted premiums for business interruption claims, using Pareto ap-proximation (with diﬀerent thresholds u). 112
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks 5 e−02 1 e−02 1−F(x) (on log scale) 1−F(x) (on log scale) 95 95 2 e−03 5 e−03 99 99 5 e−04 5 e−04 1 e−04 1 e−04 1 2 5 10 20 50 100 200 5 10 20 50 100 200 x (on log scale) x (on log scale)Figure 51: Estimation of the VaR and the TVaR with levels 99, 9%, where u = 1on the left, u = 5 on the right. 113
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks AgendaCatastrophic risks modelling • General introduction • Business interruption and very large claims • Natural catastrophes and accumulation risk • Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements • Risk measures, an economic introduction • Calculating risk measures for catastrophic risks • Diversiﬁcation and capital allocation 114
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Coherent risk measuresA risk measure is said to be coherent (from Artzner, Delbaen, Eber &Heath (1999)) if • R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ), • R(·) is positively homogeneous, i.e. for any λ ≤ 0, R(λX) = λR(X), • R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ, • R(·) is subadditive, i.e. R(X + Y ) ≤ R(X) + R(Y ).“subadditivity” can be interpreted as “diversiﬁcation does not increase risk”.Example: the Expected-Shortfall is coherent, the Value-at-Risk is not. 115
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Non-subadditivity of Value-at-RiskIn the insurance context, Ewans (2001) pointed out that “probability of ruin mayoften be inconsistent with many other reasonable risk management criteria”.Example: Consider X, Y ∼ LN (0, 1). 116
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks Possible VaR for the sum of two LN(0,1) risks 40 30 20 VaR(X+Y) > VaR(X)+VaR(Y) 10 VaR(X+Y) < VaR(X)+VaR(Y) 0 0.90 0.92 0.94 0.96 0.98 1.00 Probability levels Figure 52: Value-at-Risk for the sum of lognormal risks. 117
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Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks The non subadditivity puzzle for large risksAssume that X and Y have tail indices a and b. If min a, b > 1, there existsp0 ∈ (0, 1) such that for all p ∈ (p0 , 1), V aR (X + Y, p) < V aR(X, p) + V aR(Y, p).If min a, b < 1, there exists p0 ∈ (0, 1) such that for all p ∈ (p0 , 1), V aR (X + Y, p) > V aR(X, p) + V aR(Y, p). 118
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